Sar Image Formation

7/28/2019 Sar Image Formation

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SAR IMAGE FORMATION:

ERS SAR PROCESSOR CODED IN MATLAB(Copyright 2002, David T. Sandwell)

The Range-Doppler Algorithm (Curlander and McDonough, Synthetic Aperture Radar:Systems & Signal Processing, Chapter 4, John Wiley & Sons, New York, 1991.)

The basic SAR theory is conceptually simple yet when one looks into the inner workings of the

SAR processor it appears quite complicated. The most difficult part of the problem is related tothe coupling between the azimuthal compression and the orbital parameters. In the old days

when the satellite orbits were not known very accurately, one would use techniques such as

clutterlockand autofocus to derive the orbital parameters from the data. This is no longer

necessary with the high accuracy orbits available today. A standard processing sequence followswhere the first two steps are done onboard the satellite while the remaining 4 steps are done by

the user with a digital SAR processor.

Processing Onboard the Satellite

Demodulate - The electromagnetic wave consists of a chirp of bandwidth B (~15 MHz)

superimposed on carrier frequency o (~5 GHz). To record the return signal, one could simply

digitize the amplitude of the electric field E(t) at a sampling rate of 10 GHz using 1-byte

encoding. However, this would generate 10 Gbytes of data every second! We are really only

interested in how the chirp has been delayed and distorted by the reflection off the Earthsurface. The trick here is to use the shift theorem [Bracewell, 1978, p. 108] to isolate the 15

MHz part of the spectrum containing the chirp. Suppose we multiply the signalE(t) by e i2ot.

This will shift the fourier transform of the signal as shown in the diagram below.

The functionE(t) is a real-valued function so we know that its fourier transform has Hermitiansymmetry about the origin, that is E( -)=E*(). Because of this symmetry, all of the

information in the time series is contained within the bandwidth of a single spectral peak. Next

we low-pass filter the signal to isolate the spectral peak centered around the zero frequency.

The original signalE(t) was a real-valued function but after this shift/filter operation the output

complex. No information is lost in this process and now the output can be digitized at the muchlower rate of twice the bandwidth of the chirp (~30 MHz).

-o 0 o 2o


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Digitize The complex signal is digitized at 5 bits per pixel so the numbers range from 0 to 31;

the mean value is about 15.5. Digitizing the data at 8 bits per pixel would seem more convenientbut sending these extra 3 bits to the ground receiving exceeds the bandwidth of the telemetry so

every effort is made to compress the data before transmission. Once the data are on the ground

they are expanded to 8 bits (one byte) for programming convenience. Engineers refer to the real

and imaginary parts of the signal as the in-phase (I) and quadrature (Q) components. The rawsignal files contain rows of (11644 bytes) that represent a single radar echo. The first 412 bytes

are timing and other information while the remaining 11232 bytes contain the 5616 complex

numbers of raw signal data.

Digital SAR Processing

The digital SAR processor is a computer program that converts the raw signal data into a single-

look complex (SLC) image. An overview is provided in the diagram below this is followed by adetailed description of each step. With this information one can write a basic SAR processor

using just a few lines of code in MATLAB. An example program is provided in the Appendix.


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Digital SAR Processing Overview

(1)Range compression performed on each radar echo

(2)Patch processing of range-compressed echos

(3)

Range migration performed on each pixel of range-compresseddata whose columns have been fourier transformed

(4)

Azimuth compression performed on each column of range-compressed, range-migrated data

Parameter

file

Single Look

Complex (SLC)image data file

Raw signal

data file


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Raw Signal Data

ran e

azimuth

patch

incomplete aperture

complete aperture

incomplete aperture

412 bytes 11232 bytes 5161 complex numberstiming raw signal data


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Parameter File (e2_10001_2925.PRM)

file information/format

input_file e2_10001_2925.fix name of raw data file

SC_identity 2 (1) ERS-1 or (2) ERS-2

bytes_per_line 11644 number of bytes in row of raw datafirst_sample 206 412 bytes of timing information to skip over

fd1 248.115 Doppler centroid estimated from data fDcI_mean 15.504000 mean value of real numbers

Q_mean 15.549000 mean value of imaginary numbersicu_start 2576039268.848 spacecraft clock for first sample

SC_clock_start 1997078.767881851 UTC for first sample YYYDDD.DDDDDDDD

SC_clock_stop 1997078.768074699 UTC for last sample YYYDDD.DDDDDDDD

radar characteristics

PRF 1679.902394 pulse repitition frequency PRF

rng_samp_rate 1.89625e+07 range sampling rate fschirp_slope 4.17788e+11 chirp slope k

pulse_dur 3.712e-05 pulse duration pradar_wavelength 0.056666 radar wavelength

orbital information

near_range 829924.365777 distance to first range bin Rnearearth_radius 6371746.4379 earth radius 1/2 way into image ReSC_height 787955.52 spacecraft height 1/2 way into image H

SC_vel 7125.0330 ground-track velocity V

processing informationnum_valid_az 2800 size of patch to processnum_patches 10 number of patches to process

first_line 1

deskew n deskew (yes or no)

st_rng_bin 1 start processing at row numbernum_rng_bins 6144 number of range bins in output file

chirp_ext 614 extend the chirp to process outside swath

Flip_iq n exchange real and imaginary numbers (yes or no)nlooks 1 number of azimuth echoes to average

image alignmentrshift 15.1 range shift to align image to masterashift 483.2 azimuth shift to align image to master

stretch_r .0014569 range stretch versus range

stretch_a -.0019436 azimuth stretch versus range

a_stretch_r 0.0 range stretch versus azimutha_stretch_a 0.0 azimuth stretch versus azimuth


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(1)Range Compression A sharp radar pulse is recovered by deconvolution of the chirp.This is done with fast fourier transform. There are 5616 points in the ERS-1 signal dataand the chirp is 703 points long. Both are zero-padded to a length of 8192 prior to

deconvolution to take advantage of the speed radix-2 fft algorithms. (Later we keep 6144

points, which enables one to extract the phase beyond the edges of the original swath.

Since the chirp really did interact with the ground outside the digitized swath, this is notmagic. The number 6144 has small prime factors that may be optimal for later 2-D fft

phase unwrapping algorithms.)

(2)Patch Processing The next step is to focus the image in the along-track or azimuthdirection. This is also done by fft, but now we need to process columns rather than rows.

For the ERS radar, the synthetic aperture is 1296 points long so we need to read at least

that many rows into the memory of the computer. Actually 4096 rows is better number

since it is a power of 2. A typical ERS raw data file has 28,000 rows so we will need toprocess many of these patches. One problem with this approach is that the synthetic

aperture is only complete for the inner 2800 points of the 4096 data patch so the patches

must have a 1296-point overlap. A 4096 x 5616 patch requires at least 184 Mbytes ofcomputer memory for single precision real numbers. In the old days before this largememory space was available, each patch was transposed - corner turning using high-

speed disks so the data could be processed on a column-by-column basis. Nowadays a

multiprocessor computer could process many patches in parallel and assemble the output

files in the proper order since the overlapping patches are independent.

(3)Range Migration As we will see below, a point target will appear as a hyperbolic-shaped reflection as it moves through the synthetic aperture. In addition, there could be apronounced linear drift due to an elliptical orbit and earth rotation. In other words, the

target will migrate in range cell as a linear trend plus a hyperbola. The shape of this

migration path is calculated from the precise orbital information. Prior to focusing theimage along a single column, these signals must be migrated back to a constant rangecell. This is called range migration and the fastest way to do this is by fourier

transforming the columns first. Each fourier component corresponds to a unique Doppler

shift and also a unique value of range migration; For an ideal radar, the first component

will have zero Doppler and will correspond to the point on the earth that is perpendicularto the spacecraft velocity vector. The second component will have a small positive

Doppler shift and a small migration of the range cells will be needed - and so on all the

way throught the positive and negative Doppler spectrum.

(4)Azimuth Compression The final step in the processing is to focus the data in azimuth byaccounting for the phase shift of the target as it moves through the aperture. In thelecture on diffraction, we calculated the illumination pattern on a screen due to the

propagation of a coherent wave from the aperture to the screen. The azimuthcompression relies on the same theory although we do the opposite calculation given

the illumination pattern, calculate the shape of the aperture (reflectivity of the target).

This is done by generating a second frequency-modulated chirp where the chirp

parameters depend on the velocity of the spacecraft, the pulse repetition frequency (PRF),and the absolute range. The chirp is fourier transformed into Doppler space and


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multiplied by each column of range-migrated data. The product is inverse fourier

transformed to provide the focused image.

Single-Look Complex (SLC) Image As described above, only the inner 2800 rows use

the complete synthetic aperture so these are written to an output file. The file-reading

pointer is moved back 1296 rows to start the processing of the next patch. Becausecomplete apertures are used for each patch, they will abut seamlessly.

Range Compression

To reduce the peak power of the radar transmitter associated with a short pulse, a longfrequency-modulated chirp is emitted by the radar. This chirp propagates to the ground where it

reflects from a swath typically 100 km wide. When it returns to the radar, the raw signal data

consists of the complex reflectivity of the surface convolved with the chirp. Our objective is to

recover the complex reflectivity by deconvolution of the chirp. As discussed above, this is thefirst step in the digital SAR processor. For the ERS-radar, the frequency-modulated chirp is

s(t) = eikt

2

t < p /2 .

wherek - chirp slope (4.17788 x 1011 s-2)

p - pulse duration (3.712 x 105 s or ~11 km long)

fs - range sampling rate (1.89625 x 107 s-1).

An example of a portion of the chirp for the ERS- radar as well as its power spectrum and

impulse response is shown below.


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The bandwidth of the chirp, B = kp , has a value of 15.5 MHz for the ERS radar. A matched

filter is used to deconvolve the chirp from the data. In this case, the matched filter is simply the

complex conjugate of the chirp or s*(t) = eikt

2

. The convolution ofs*(t) and s(t) is shown in the

figure above. The effective range resolution of the radar is about 27 m.

Using MATLAB code (Appendix) and the radar parmaters provided above, one can deconvolve

the ERS chirp. The figure below is a small patch of ERS-2 data for an area around Pinyon Flat,

California (track 127, frame 2925, orbit 10001). This area contains two radar reflectors (seefigure below) spaced about 600 m apart that were installed in late 1996. The amplitude of the

raw signal data appears as noise with a single horizontal streak due to missing data. The range-

compressed data (right image) shows two vertical streaks due to the high reflectivity of thereflectors (red circles) as they migrate through the synthetic aperture.


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Azimuth Compression

Azimuth compression or azimuth focusing involves generation of a frequency-modulated chirpin azimuth based on the knowledge of the spacecraft orbit. The geometry of the strip-mode

acquisition is shown below.

s slow time along the satellite track

x ground-track positionV ground track velocity

so time when the target is in the center of the radar illumination pattern

H spacecraft height

Rg ground rangeR(s) range from spacecraft to target

Ro = H2

+ Rg2

= Rnear + n *(C/ fs) minimum range from the spacacraft to the target

The complex phase of the return echo is

C(s) = exp i4

R(s)

s

Rg

(x-sV)

R(s)


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where the range is

R2(s) = H

2+ Rg

2+ x sV( )

2.

This is a hyperbola that we can approximate using a parabola

R(s) =Ro+

Ro(s s

o)+

Ro

2(s s

o)2+ ...

where the dot indicates derivative with respect to slow time, s. Curlander and McDonough

[1991] discuss the accuracy of this polynomial approximation. Is is good enough for strip-modeSAR but may be inadequate for the much longer apertures associated with spotlight-mode SAR.

Now we need to calculate R and R in terms of spacecraft parameters. Lets start with R bytaking the derivative ofR2 with respect to s.

R2

s= 2R R R =

1

2R

R2

s

We note that Ro

is the velocity of the spacecraft in the range direction at the time so

and thus a

measure of the Doppler shift when the target is in the center of the radar illumination pattern.

Ro= V

x soV( )

Ro

The range acceleration R is the derivative of the range rate R.

R =

s

1

2R

R2

s

=

1

2

1

R

2R

2

s2

+

1

R2

R

s

R2

s

This can be written as

R =V

2

R+

V2 x sV( )2

R3

=

V2

R1

x sV( )2

R2

.

For the orbital characteristics of the ERS spacecraft, the second term is small so we have

RoV

2

Ro

Now we can write the phase of the return signal as a function of geometry and speed


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C(s) = exp i4

R

o

2+

Ros s

o( )+ Ro s so( )2/2[ ]

or

C(s) = exp i

4Ro

2

exp i22V

x soV( )

Ro

s so( )+2V2

Ro

s so( )2

/2

Note that this function is another frequency-modulated chirp where and there are two important

parameters theDoppler centroid

fDC =2V

x soV( )R

o

and theDoppler frequency rate

fR =2V

2

Ro.

An example of this azimuthal chirp function for the ERS orbit/radar as well as its powerspectrum and impulse response are shown below.

Dopplercentroid

Doppler

frequencyrate


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The bandwidth of the azimuthal chirp is equal to the pulse repition frequency (PRF = 1680 Hz).

In this case the matched filter is simply the complex conjugate of the azimuthal phase function orC*(s).

Azimuthal Compression Parameters and ERS Orbit

The Doppler centroid and Doppler frequency rate depend on the ground velocity of thespacecraft V, the wavelength of the radar , the minimum distance between the spacecraft and

the target Ro, and a factor

x soV

Ro

which is the squint angle.

Ground velocity - First consider the ground velocity of the spacecraft. Precise orbital

information is used to compute the velocity of the spacecraft at satellite altitude Vs. As anexercise, show that the ground velocity is

V =V

s

1+H

Re

1/ 2

whereHis the local spacecraft height andReis the local earth radius.

Range - The quantity Ro

is the range to a particular column in the raw signal data file.

Ro = Rnear + n(c /2 fs)

where Rnear

is the near range to the first column of the raw signal data file, n is the column

number (0-5615), c is the speed of light andfs is the range sampling rate given in the previoussection on range compression.

Squint angle - Finally lets examine the quantity,x s

oV

Ro

. This is related to something called the

squint angle, ,as shown in the following diagram.

s

Ro

- soV


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The squint angle is simply

= tan1 x soV

Ro

.

For the ERS satellites, the squint angle is adjusted so the zero Doppler occurs roughly in the

center of the antenna beam pattern. If the satellite orbit was circular and the earth was a non-

rotating sphere, then the zero Doppler would correspond to a zero squint angle.

For interferometry it is important that the beam patterns of the reference and repeat orbits have a

large overlap at a given along-track spacecraft position. The requirement is that the Doppler

centroid of the reference and repeat passes agree to about 1/2 of the pulse repetition frequency

(PRF). In late 1999, the gyroscopes on ERS-2 failed so it became difficult to control the squint

angle of the spacecraft. Data acquired after this date may a Doppler centroid outside of theacceptable range and thus may be useless for interferometry.

Using MATLAB code (Appendix) and the azimuth compression provided above, one canfocusthe image in the azimuth direction. The figures below the same small patch of ERS-2 data for an

area of Pinyon Flat, California. The range-compressed data (left image) shows two vertical

streaks due to the high reflectivity of the reflectors (red circles) as they migrate through the

synthetic aperture. The fully focused image (right) shows the high point-like reflectivityassociated with the two radar reflectors.


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Appendix MATLAB Code for Focusing ERS Signal Data

%****************************************************

function [cref,fcref]=rng_ref(nfft,fs,pulsedur,slope)

%

% routine to compute ERS chirp and its fourier

transform

%

% input

% fs - sampling frequency, ts=1./fs

% pulsedur - pulse duration

% slope - chirp slope

%

% set the constants and make npts be odd

%

npts=floor(fs*pulsedur);

ts=1./fs;

if(mod(npts,2.0) == 0.0)

npts=npts+1;

end

%

% compute the reference function%

npt2=floor(npts/2.);

t=ts*(-npt2:npt2);

phase=pi*slope*t.*t;

cref1=exp(i*phase);

%

% pad the reference function to nfft

%

cref=[cref1,zeros(1,nfft-npts)]';

%

% compute the fourier transform

%

fcref=fft(cref)/nfft;

%****************************************************

function [cazi,fcazi]=azi_ref(nazi,PRF,fdc,fr)

%

% routine to compute ERS azimuthal chirp and its

fourier transform

%

% input

% nazi - number of points in azimuth

% PRF - pulse repitition frequency, ts=1./fs

% fdc - doppler centriod frequency

% fr - doppler frequency rate

%

% set the constants and make npts be odd

%npts=min(nazi-1,1296);

ts=1./PRF;

if(mod(npts,2.0) == 0.0)

npts=npts+1;

end

%

% compute the azimuth chirp function

%

npt2=floor(npts/2.);

t=ts*(-npt2:npt2);

phase=-2.*pi*fdc*t+pi*fr*t.*t;

cazi1=exp(i*phase);

%

% pad the function to nfft

%

cazi=[cazi1(npt2:npts),zeros(1,nazi-

npts),cazi1(1:npt2-1)]';

%

% compute the fourier transform

%

fcazi=fft(cazi)/nazi;

%****************************************************

function [csar,nrow,ncol]=read_rawsar(sar_file)

%

% routine to read and unpack ERS SAR data in

DPAF format

%fid=fopen(sar_file,'r');

%

% read the bytes

%

[sar,nsar]=fread(fid,'uchar');

sar=reshape(sar,11644,nsar/11644);

nrow=nsar/11644;

ncol=5616;

st=fclose(fid);

%

% extract the real and imaginary parts

% and remove the mean value

%

csar=complex(sar(413:2:11643,:)-15.5,sar(414:2:11644,:)-15.5);

%

%****************************************************

% matlab script to focus ERS-2 signal data

%

% set some constants for e2_10001_2925

%

% range parameters

%

rng_samp_rate = 1.896e+07;

pulse_dur = 3.71e-05;

chirp_slope = 4.1779e+11;

%% azimuth parameters

%

PRF=1679.902394;

radar_wavelength=0.0566666;

SC_vel=7125.;

%

% compute the range to the radar reflectors

%

near_range=829924.366;


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dr=3.e08/(2.*rng_samp_rate);

range=near_range+2700*dr;

%

% use the doppler centroid estimated from the data

and the

% doppler rate from the spacecraft velocity and

range

%fdc=284;

fr=2*SC_vel*SC_vel/(range*radar_wavelength);

%

% get some sar data

%

[cdata,nrow,ncol] = read_rawsar('rawsar.raw');

%

% make a colormap

%

map=ones(21,3);

for k=1:21;

level=0.05*(k+8);

level=min(level,1);

map(k,:)=map(k,:).*level;end

colormap(map);

%

% image the raw data

%

figure(1)

subplot(2,2,1),imagesc(abs(cdata'));

xlabel('range')

ylabel('azimuth')

title('unfocussed raw data')

axis([2600,2900,1000,1200])

%

% generate the range reference function

%

[cref,fcref]=rng_ref(ncol,rng_samp_rate,pulse_dur,chir

p_slope);

%

% take the fft of the SAR data

%

fcdata=fft(cdata);

%

% multiply by the range reference function

%

cout=0.*fcdata;

for k=1:nrow;

ctmp=fcdata(:,k);

ctmp=fcref.*ctmp;cout(:,k)=ctmp;

end

clear cdata

%

% now take the inverse fft

%

odata=ifft(cout);

clear cout

%

% plot the image and the reflector locations

%

x0=[2653.5,2621];

x0=x0+90;

y0=[20122,20226];

y0=y0-19500+427;

figure(1)hold

subplot(2,2,2),imagesc(abs(odata'));

plot(x0,y0,'o')

xlabel('range')

ylabel('azimuth')

title('range compressed')

axis([2600,2900,1000,1200])

%

% use this for figure 2 as well

%

figure(2)

colormap(map);

subplot(2,2,1),imagesc(abs(odata'));

hold onplot(x0,y0,'o')

xlabel('range')

ylabel('azimuth')

title('range compressed')

axis([2600,2900,1000,1200])

%

% generate the azimuth reference function

%

[cazi,fcazi]=azi_ref(nrow,PRF,fdc,fr);

%

% take the column-wise fft of the range-compressed

data

%

fcdata=fft(odata');%

% multiply by the azimuth reference function

%

cout=0.*fcdata;

for k=1:ncol;

ctmp=fcdata(:,k);

ctmp=fcazi.*ctmp;

cout(:,k)=ctmp;

end

%

% now take the inverse fft and plot the data

%

odata=ifft(cout);

figure(2)subplot(2,2,2),imagesc(abs(odata));

hold on

plot(x0,y0,'o')

xlabel('range')

ylabel('azimuth')

title('range and azimuth compressed')

axis([2600,2900,1000,1200])

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